20 problems

Let a Markov chain have $n$ states, let $x$ and $y$ be states, and let $\tau_y$ be the hitting time of $y$ starting from $x$. For times $t>n$, consider the interval probability…

Let $G$ be the evolving random graph process, and write $\tau(\mathcal E)$ for the hitting time at which event $\mathcal E$ first occurs. Let ${\cal M}$ be the perfect matching gam…

Let $G$ be a finite graph with $n$ vertices, let $x,y eads?$ be vertices, and let $tauy$ denote the hitting time of $y$ for simple random walk started at $x$. The surprise probabil…

Let $\tau(\alpha,\rho)$ be the first hitting time of zero for a real strictly $\alpha$-stable process starting from one, where $\alpha\in(1,2]$ and $\rho$ is the positivity paramet…

Let $P_{-M}$ be the transition matrix restricted to the non-marked vertices, and let $\mathcal{D}_{-M}D_r^{1/2}=V\Lambda V^{-1}$, where…

Let $n\equiv 1,3 \pmod 6$, and let $T_1,T_2,\ldots$ be a uniformly random ordering of the triples in $\binom{[n]}{3}$. Steiner triple system hitting-time conjecture. With high prob…

Distributional-limit conjecture. If $\kappa\in(2,3)$, then

Let $G=(V,E)$ be a bounded-degree, $d$-regular graph with $n=|V|$ vertices. For a subset $A\subseteq V$, write $A^c=V\setminus A$ and let $|E(A,A^c)|$ denote the number of edges cr…

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is…

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, and expose its edges one by one in a uniformly random order to obtain the random hypergraph process. A facet is…

Let $T_p:[0,1]\to[0,1]$ be the interval map with an indifferent fixed point at $x^*=0$ described above, where $p\in(0,1)$. For shrinking neighbourhoods $[0,\epsilon]$ of $x^*$, let…

Assume $E[\log\rho_0]<0$, and assume the stronger definition of $\kappa$ together with $E[\rho_0^\kappa\log\rho_0]<\infty$ and a non-lattice distribution for $\log\rho_0$, as descr…

Consider a nearest-neighbor open quantum random walk on the vertices $\{0,\dots,M\}$ induced by matrices $L$ and $R$ satisfying … For an initial internal density matrix $\rho$, let…

Lollipop conjecture. Graphs attaining the maximum value of the average hitting time correspond to a certain type of lollipop graph.

Chen et al.'s conjecture. The expected hitting time in the general $n$-urn case should be

Let $X$ be a reversible Markov chain with finite state space, and let $\pi$ be the stationary distribution of $X$. For states $x$ and $y$, write $\tau(y)$ for the hitting time of…

Greater-generality conjecture. The estimates for $\mathbf{P}(\tau_x>t)$ established under the scaling-type condition should remain valid under this weaker condition.

Connectivity-threshold conjecture. The results established for denser Erdős–Rényi graphs, in particular $H_j=N+o(N)$, are true already when

Peres's conjecture. In 2009 Peres conjectured that ${\rm T}_{\rm hit}^Q(1/2)$ is equivalent up to universal constant factors to

Let $\{X(t)\}$ be an ergodic Markov chain on $\mathbb{N}$ with stationary distribution $\pi$. For $[n]=\{1,\ldots,n\}$, let $T_n>0$ be a hitting time for $[n]^c$, and let…